In case these problems become more complex in the case that writing them out is not very efficient, note that you can still use your old rules of differentiation with matrices and vectors, but since matrices and vectors are not necessarily commutative, you have to remember the order of multiplication. That is if you have Ax + xA, it is not okay to write that it is 2Ax.
In respect to your original question, lets look at the product of three matrices at the end:
Applying the product rule, we get:
The proof for this is just using vectors and matrices in place of numbers in the usually proofs of these properties; the complication is that the limits for vectors and matrices are more difficult to verify, as the vanishing component is also a vector/matrix.

Most definitely. However, when used over time, the general laws make problems easier. Ie., one doesn't usually write all of that out on paper; one can jump immediately from the first line to the penultimate one. I only went through all the intermediate steps so you need to see how the normal laws of differentiation are being applied.